Any route traveling from (0,0) to (1,1) going only north and east will
cover a total distance of 2 units. But the straight line distance from
(0,0) to (1,1) is sqrt(2) units. It seems that if I think of a
staircase connecting the two points and let the stairs become
infinitely small, the limit of the north/east route distance should
converge to sqrt(2). But it doesn't! What's going on here?

Can you help me show, with and without calculus, that the geometric figure of a maximum area and given perimeter is a circle? What are the dimensions of a triangle with perimeter p that encloses the maximum area?

Two observers on points A and B of a national park see a beginning fire
on point C. Knowing that the angles CAB = 45 degrees, ABC = 105 degrees
and that the distance between points A and B is of 15 kilometers,
determine the distances between B and C, and between A and C.

We live in an area known as the Research Triangle, with the triangle's
points at the University of North Carolina, North Carolina State
University and Duke University. We are interested in finding the center
point of our triangle home and whether there is a unique term (or several
terms) for the center point of a triangle.

How can I find the coordinates of the point A of triangle ABC if B lies
on the line 3y = 4x, C lies on the line y = 0, the line BC passes through
(2/3,2/3) and AOBC forms a rhombus (where O is the origin)?

I need to find the total square footage of a lot of rectangular lawns.
Do I have to find the area of each lawn and add up all the areas, or
can I just add all the lengths and all the widths and make one area
calculation based on those two totals?

Let ABC be a right-angled triangle with angle C = 90 degrees. Let
the bisectors of angle A and angle B intersect BC and CA at D and E
respectively. Given that CD = 9 and CE = 8, find the lengths of the
sides of ABC.